Genus 0 curves are well understood in number theory. There is also are rich theory a bunch of conjectures about the arithmetic of elliptic curves.

This leads me to the question, what we know about higher genus curves.
The only thing that comes to my mind is Falting's theorem saying that
any such curve has only finitely many points.

Coming from the theory of elliptic curves one might ask a lot of questions:

Under BSD there is an algorithm computing a basis for Mordell-Weil group
of the elliptic curve.
Is it expected that such an algorithm will also exist for higher genus curves that
determines all the rational points, or even proven to exists assuming some big
conjectures such as ABC or Vojta's conjectures. Or does Matiyasevich theorem
imply that such an algorithm is not possible?

Do special values of $L$-functions of these curves tell us anything interesting about
these curves. I know that there is a generalization of BSD to abelian varieties
and hence to Jacobians of higher genus curves, but does the $L$-functions also
tells something about the curve itself.

The local-global principle does not hold for elliptic curves. However, (conjecturally
through BSD) local information does indeed give information about global properties.
So I regard BSD as a fancier or refined version of local-global principle, since the
pure local-global principle does not hold. Is there also a fanicer local-global
principle for higher genus curves.

I am not particulary asking answers for these questions, since these are only examples and I could post more. I am rather asking for a place to look, where things like these are treated.

In fact, according to Milne's style page, it should be Faltings's theorem. I don't know whether this is generally accepted style or not, however.
–
EmertonApr 4 '11 at 23:54

@wood : in your point 3, do you mean that the local-global principle does not hold for genus one curves ?
–
François BrunaultApr 5 '11 at 0:08

1

definitely Faltings's; Faltings' means it was proved by multiple people named Falting.
–
Ben Webster♦Apr 5 '11 at 0:19

1

The very first line of Strunk and White's The Elements of Style is "Form the possessive singular of nouns by adding 's ". The second is: "Follow this rule whatever the final consonant. Charles's friend..."
–
Georges ElencwajgApr 5 '11 at 0:29

2 Answers
2

There is a conjecture (or variants of one) due to Scharaschkin, Skorobogatov and Stoll predicting that Brauer-Manin or finite descent obstructions (these are fancy forms of a local-global principle) determine all rational points. This gives an algorithm to determine if the set of points is empty or not. See Stoll, Algebra and Number Theory 2 (2008), 595–611. The algorithm is actually practical, as demonstrated by Bruin and Stoll (for genus two over the rationals). This answers your 3 and part of 1.

For the rest of 1, Vojta's conjecture or ABC (with some choice of constant which in the original conjectures is not specified) would give an upper bound for the height of rational points on a curve and would allow the determination of all such points by a finite search.

They develop much of the descent theory and theory of homogeneous spaces explicitly for curves of genus 2.

Concerning question (2), as Felipe suggests, the L-series of the curve contains the same information as the L-series of its Jacobian, so BSwD will tell you about the rank of the Jacobian and the height of generators for $J(K)$. In particular, if $L(1)\ne 0$, then the rank is 0, so $C(K)$ consists of torsion points, which would allow one to effectively determine $C(K)$. In general, it is reasonable to suppose that there might be a bound for the height of the largest point in $C(K)$ in terms of a generating set for $J(K)$ (although such a bound is not currently known). Then BSwD would give a bound for the latter, and hence for the former.